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Entire solutions in${\mathbb{R}}^{2}$ for a class of Allen-Cahn equations

Published online by Cambridge University Press:  15 September 2005

Francesca Alessio  and
Piero Montecchiari
Francesca Alessio
Affiliation:
Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy; alessio@dipmat.univpm.it;montecch@mta01.univpm.it
Piero Montecchiari
Affiliation:
Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy; alessio@dipmat.univpm.it;montecch@mta01.univpm.it
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Abstract

We consider a class of semilinear elliptic equations of the form 15.7cm -$\varepsilon^{2}\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in{\mathbb{R}}^{2}$ where $\varepsilon>0$, $a:{\mathbb{R}}\to{\mathbb{R}}$ is a periodic, positive function and $W:{\mathbb{R}}\to{\mathbb{R}}$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in{\mathbb{R}}$. We show via variational methods that if ε is sufficiently small and a is not constant, then ([see full textsee full text]) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

Keywords

Heteroclinic solutionselliptic equationsvariational methods.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 4 , October 2005 , pp. 633 - 672
Copyright
© EDP Sciences, SMAI, 2005

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